Let’s explore how engineers and physicists use Fourier series to model and solve real-world discontinuous periodic systems. Consider a perfect square wave—a signal that jumps instantly between +1 and -1. This is the poster child for discontinuity. Its Fourier series is:
[ E(x) = e^{i k x} \sum_{n=-\infty}^{\infty} E_n , e^{i n K x} ] Let’s explore how engineers and physicists use Fourier
[ f(x) = \frac{4}{\pi} \sum_{n=1,3,5,\ldots} \frac{\sin(nx)}{n} ] Its Fourier series is: [ E(x) = e^{i
Don’t fear the jump. Embrace the Fourier series—just remember to keep enough harmonics to capture the edge. Meanwhile, the electric field or pressure wave is
Even with jumps, the Fourier coefficients (\varepsilon_m) decay as (1/m) (for a step change). Meanwhile, the electric field or pressure wave is assumed to follow Bloch’s theorem:
[ \varepsilon(x) = \sum_{m=-\infty}^{\infty} \varepsilon_m , e^{i m K x}, \quad K = \frac{2\pi}{a} ]